The number of solutions of the equation $32 ^ { \tan ^ { 2 } x } + 32 ^ { \sec ^ { 2 } x } = 81 , \quad 0 \leq x \leq \frac { \pi } { 4 }$ is :
(1) 0
(2) 2
(3) 1
(4) 3

The number of roots of the equation, $( 81 ) ^ { \sin ^ { 2 } x } + ( 81 ) ^ { \cos ^ { 2 } x } = 30$ in the interval $[ 0 , \pi ]$ is equal to :
(1) 3
(2) 4
(3) 8
(4) 2

The number of solutions of the equation $x + 2 \tan x = \frac { \pi } { 2 }$ in the interval $[ 0,2 \pi ]$ is
(1) 3
(2) 4
(3) 2
(4) 5

Let $S$ be the sum of all solutions (in radians) of the equation $\sin ^ { 4 } \theta + \cos ^ { 4 } \theta - \sin \theta \cos \theta = 0$ in $[ 0,4 \pi ]$ then $\frac { 8 S } { \pi }$ is equal to

The number of solutions of the equation $\cos \left( x + \frac { \pi } { 3 } \right) \cos \left( \frac { \pi } { 3 } - x \right) = \frac { 1 } { 4 } \cos ^ { 2 } 2 x , x \in [ - 3 \pi , 3 \pi ]$ is:
(1) 8
(2) 5
(3) 6
(4) 7

The number of solutions of $\cos x = \sin x$, such that $- 4 \pi \leq x \leq 4 \pi$ is
(1) 4
(2) 6
(3) 8
(4) 12

Let $S = \left\{ \theta \in [ 0,2 \pi ] : 8 ^ { 2 \sin ^ { 2 } \theta } + 8 ^ { 2 \cos ^ { 2 } \theta } = 16 \right\}$. Then $n ( S ) + \sum _ { \theta \in \mathrm { S } } \left( \sec \left( \frac { \pi } { 4 } + 2 \theta \right) \operatorname { cosec } \left( \frac { \pi } { 4 } + 2 \theta \right) \right)$ is equal to:
(1) 0
(2) $- 2$
(3) $- 4$
(4) 12

Let $S = \left\{ \theta \in \left( 0 , \frac { \pi } { 2 } \right) : \sum _ { m = 1 } ^ { 9 } \sec \left( \theta + ( m - 1 ) \frac { \pi } { 6 } \right) \sec \left( \theta + \frac { m \pi } { 6 } \right) = - \frac { 8 } { \sqrt { 3 } } \right\}$. Then
(1) $\mathrm { S } = \left\{ \frac { \pi } { 12 } \right\}$
(2) $S = \left\{ \frac { 2 \pi } { 3 } \right\}$
(3) $\sum _ { \theta \in S } \theta = \frac { \pi } { 2 }$
(4) $\sum _ { \theta \in S } \theta = \frac { 3 \pi } { 4 }$

Let $S = \left[ - \pi , \frac { \pi } { 2 } \right) - \left\{ - \frac { \pi } { 2 } , - \frac { \pi } { 4 } , - \frac { 3 \pi } { 4 } , \frac { \pi } { 4 } \right\}$. Then the number of elements in the set $A = \{ \theta \in S : \tan \theta ( 1 + \sqrt { 5 } \tan ( 2 \theta ) ) = \sqrt { 5 } - \tan ( 2 \theta ) \}$ is $\_\_\_\_$.

Let $S = \{ \theta \in [ 0,2 \pi ) : \tan ( \pi \cos \theta ) + \tan ( \pi \sin \theta ) = 0 \}$, then $\sum _ { \theta \in S } \sin ^ { 2 } \left( \theta + \frac { \pi } { 4 } \right)$ is equal to

Let $S = \left\{ x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) : 9 ^ { 1 - \tan ^ { 2 } x } + 9 ^ { \tan ^ { 2 } x } = 10 \right\}$ and $\beta = \sum _ { x \in S } \tan ^ { 2 } \frac { x } { 3 }$, then $\frac { 1 } { 6 } ( \beta - 14 ) ^ { 2 }$ is equal to
(1) 16
(2) 8
(3) 64
(4) 32

Let $S = \{\theta \in [0, 2\pi): \tan(\pi\cos\theta) + \tan(\pi\sin\theta) = 0\}$. Then $\sum_{\theta \in S} \sin\left(\theta + \frac{\pi}{4}\right)$ is equal to $\_\_\_\_$.

If $m$ and $n$ respectively are the numbers of positive and negative value of $\theta$ in the interval $[ - \pi , \pi ]$ that satisfy the equation $\cos 2 \theta \cos \frac { \theta } { 2 } = \cos 3 \theta \cos \frac { 9 \theta } { 2 }$, then $m n$ is equal to $\_\_\_\_$.

The set of all values of $\lambda$ for which the equation $\cos ^ { 2 } 2 x - 2 \sin ^ { 4 } x - 2 \cos ^ { 2 } x = \lambda$ has a solution is: (1) $[ - 2 , - 1 ]$ (2) $\left[ - 2 , - \frac { 3 } { 2 } \right]$ (3) $\left[ - 1 , - \frac { 1 } { 2 } \right]$ (4) $\left[ - \frac { 3 } { 2 } , - 1 \right]$

The number of solutions of the equation $4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0$; $x \in [-2\pi, 2\pi]$ is:
(1) 1
(2) 3
(3) 2
(4) 0

Let the set of all $a \in R$ such that the equation $\cos 2 x + a \sin x = 2 a - 7$ has a solution be $[ p , q ]$ and $r = \tan 9 ^ { \circ } - \tan 27 ^ { \circ } - \frac { 1 } { \cot 63 ^ { \circ } } + \tan 81 ^ { \circ }$, then $p q r$ is equal to $\_\_\_\_$.

If $\alpha , - \frac { \pi } { 2 } < \alpha < \frac { \pi } { 2 }$ is the solution of $4 \cos \theta + 5 \sin \theta = 1$, then the value of $\tan \alpha$ is
(1) $\frac { 10 - \sqrt { 10 } } { 6 }$
(2) $\frac { 10 - \sqrt { 10 } } { 12 }$
(3) $\frac { \sqrt { 10 } - 10 } { 12 }$
(4) $\frac { \sqrt { 10 } - 10 } { 6 }$

If $2 \tan ^ { 2 } \theta - 5 \sec \theta = 1$ has exactly 7 solutions in the interval $0 , \frac { \mathrm { n } \pi } { 2 }$, for the least value of $\mathrm { n } \in \mathrm { N }$ then $\sum _ { \mathrm { k } = 1 } ^ { \mathrm { n } } \frac { \mathrm { k } } { 2 ^ { \mathrm { k } } }$ is equal to :
(1) $\frac { 1 } { 2 ^ { 15 } } 2 ^ { 14 } - 14$
(2) $\frac { 1 } { 2 _ { 1 } ^ { 14 } } 2 ^ { 15 } - 15$
(3) $1 - \frac { 15 } { 2 ^ { 13 } }$
(4) $\frac { 1 } { 2 ^ { 13 } } 2 ^ { 14 } - 15$

The number of solutions of $\sin ^ { 2 } x + \left( 2 + 2 x - x ^ { 2 } \right) \sin x - 3 ( x - 1 ) ^ { 2 } = 0$, where $- \pi \leq x \leq \pi$, is $\_\_\_\_$

The sum of all values of $\theta \in [ 0,2 \pi ]$ satisfying $2 \sin ^ { 2 } \theta = \cos 2 \theta$ and $2 \cos ^ { 2 } \theta = 3 \sin \theta$ is
(1) $4 \pi$
(2) $\frac { 5 \pi } { 6 }$
(3) $\pi$
(4) $\frac { \pi } { 2 }$

Consider the following two equations in $x$
$$\sin 2x + a\cos x = 0 \tag{1}$$ $$\cos 2x + a\sin x = -2 \tag{2}$$
over the interval $-\frac{\pi}{2} < x < \frac{\pi}{2}$, where $a > 0$.
Let $a = \sqrt{2}$. Then the value of $x$ which satisfies (1) is
$$x = \frac{\mathbf{AB}}{\mathbf{A}}$$
However, at this $x$ the value of the left side of (2) is $\mathbf{DE}$, and so equation (2) does not hold. Hence, when $a = \sqrt{2}$, (1) and (2) have no common solution.
Now, let us find a value of $a$ such that (1) and (2) have a common solution, and also the common solution $x$.
First, from (1) we have
$$\sin x = \frac{\mathbf{FG}}{\mathbf{H}}a, \quad \cos 2x = \mathbf{I} - \frac{a^2}{\mathbf{J}}.$$
When we substitute these into (2), we obtain
$$a^2 = \mathbf{K}.$$
Thus $a = \sqrt{\mathbf{K}}$, and the common solution is
$$x = \frac{\mathbf{LM}}{\mathbf{N}}$$

9. Which of the following equations have real solutions?
(1) $x^{3} + x - 1 = 0$
(2) $2^{x} + 2^{-x} = 0$
(3) $\log_{2} x + \log_{x} 2 = 1$
(4) $\sin x + \cos 2x = 3$
(5) $4 \sin x + 3 \cos x = \frac{9}{2}$

How many real numbers $x$ satisfy $\sin\left(x + \frac{\pi}{6}\right) = \sin x + \sin\frac{\pi}{6}$ and $0 \leq x < 2\pi$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5 or more

On the coordinate plane, the graph of the function $y = \sin x$ is symmetric about $x = \frac{\pi}{2}$, as shown in the figure. Find the value of $\theta$ in the range $0 < \theta \leq \pi$ that satisfies $\sin \theta = \sin\left(\theta + \frac{\pi}{5}\right)$.
(1) $\frac{\pi}{5}$
(2) $\frac{2\pi}{5}$
(3) $\frac{3\pi}{5}$
(4) $\frac{4\pi}{5}$
(5) $\pi$

What is the value, in radians, of the largest angle $x$ in the range $0 \leq x \leq 2 \pi$ that satisfies the equation $8 \sin ^ { 2 } x + 4 \cos ^ { 2 } x = 7$ ?